Question

Guest

解法を教えていただきたいです。

sinl2x)

 

1+ sin[x)

dX_

積分

Answer 1

✨ Best Answer ✨

LUX SIT

(1+sin(x))'=cosxに気付くことです.
***
sin(2x)/(1+sin(x))=2sin(x)cos(x)/(1+sin(x))=2cos(x)-{2cos(x)/(1+sin(x))}
と変形できる. これを利用すると
∫[0->π/2]{sin(2x)/(1+sin(x))}dx
=2∫[0->π/2]cos(x)dx-2∫[0->π/2]{(1+sin(x))'/(1+sin(x))}dx
=2sin(x)|[0->π/2]-2log(1+sin(x))|[0->π/2]
=2(1-log2).